Nonlinear model reduction for transport-dominated problems

2026-02-01

AI Summary

This work addresses the challenge posed by the Kolmogorov barrier, which limits the effectiveness of traditional linear reduced-order modeling approaches for transport-dominated problems—such as those involving wave propagation and moving coherent structures. To overcome this limitation, the paper proposes a unified framework that systematically categorizes and integrates existing nonlinear model reduction techniques into three classes: transformation-based mappings, online adaptive mechanisms, and strategies combining general nonlinear parameterizations with instantaneous residual minimization. By incorporating nonlinear parameterizations and dynamic adaptation, the resulting reduced-order models circumvent the constraints of linear subspaces, achieving both high accuracy and substantial computational efficiency. This framework establishes a general and effective paradigm for reduced-order modeling of transport-dominated systems.

Technology Category

• Machine Learning: Transfer, Domain Adaptation, Multi-Task Learning
• Search and Optimization: Non-convex Optimization
• Application Domains: Transportation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This article surveys nonlinear model reduction methods that remain effective in regimes where linear reduced-space approximations are intrinsically inefficient, such as transport-dominated problems with wave-like phenomena and moving coherent structures, which are commonly associated with the Kolmogorov barrier. The article organizes nonlinear model reduction techniques around three key elements -- nonlinear parametrizations, reduced dynamics, and online solvers -- and categorizes existing approaches into transformation-based methods, online adaptive techniques, and formulations that combine generic nonlinear parametrizations with instantaneous residual minimization.

Problem

• nonlinear model reduction
• transport-dominated problems
• Kolmogorov barrier
• wave-like phenomena
• coherent structures

Innovation

• nonlinear model reduction
• transport-dominated problems
• Kolmogorov barrier
• nonlinear parametrization
• residual minimization